A poroelastic finite element formulation including transport and swelling in soft tissue structures

Biomechanics of Collective Cell Migration in Cancer Progression: Experimental and Computational Methods

Cell migration is essential for regulating many biological processes in physiological or pathological conditions, including embryonic development and cancer invasion. In vitro and in silico studies suggest that collective cell migration is associated with some biomechanical particularities such as restructuring of extracellular matrix (ECM), stress and force distribution profiles, and reorganization of the cytoskeleton. Therefore, the phenomenon could be understood by an in-depth study of cells' behavior determinants, including but not limited to mechanical cues from the environment and from fellow "travelers". This review article aims to cover the recent development of experimental and computational methods for studying the biomechanics of collective cell migration during cancer progression and invasion. We also summarized the tested hypotheses regarding the mechanism underlying collective cell migration enabled by these methods. Together, the paper enables a broad overview on the methods and tools currently available to unravel the biophysical mechanisms pertinent to cell collective migration as well as providing perspectives on future development toward eventually deciphering the key mechanisms behind the most lethal feature of cancer.

An analytical poroelastic model of a non-homogeneous medium under creep compression for ultrasound poroelastography applications - Part I

An analytical theory for the unconfined creep behavior of a cylindrical inclusion (simulating a soft tissue tumor) embedded in a cylindrical background sample (simulating normal tissue) is presented and analyzed in this paper. Both the inclusion and the background are considered as fluid-filled, porous materials, each of them being characterized by a set of mechanical properties. Specifically, in this paper, the inclusion is considered to be less permeable than the background. The cylindrical sample is compressed using a constant pressure within two frictionless plates and is allowed to expand in an unconfined way along the radial direction. Analytical expressions for the effective Poisson's ratio (EPR) and fluid pressure inside and outside the inclusion are derived and analyzed. The theoretical results are validated using finite element models (FEM). Statistical analysis shows excellent agreement between the results obtained from the developed model and the results from FEM. Thus the developed theoretical model can be used in medical imaging modalities such as ultrasound poroelastography to extract the mechanical parameters of tissues and/or to better understand the impact of the different mechanical parameters on the estimated displacements, strains, stresses and fluid pressure inside a tumor and in the surrounding tissue.

FEBio: History and Advances

The principal goal of the FEBio project is to provide an advanced finite element tool for the biomechanics and biophysics communities that allows researchers to model mechanics, transport, and electrokinetic phenomena for biological systems accurately and efficiently. In addition, because FEBio is geared toward the research community, the code is designed such that new features can be added easily, thus making it an ideal tool for testing novel computational methods. Finally, because the success of a code is determined by its user base, integral goals of the FEBio project have been to offer support and outreach to our community; to provide mechanisms for dissemination of results, models, and data; and to encourage interaction between users. This review presents the history of the FEBio project, from its initial developments through its current funding period. We also present a glimpse into the future of FEBio.

Investigation of Cell-Substrate Adhesion Properties of Living Chondrocyte by Measuring Adhesive Shear Force and Detachment Using AFM and Inverse FEA

It is well-known that cell adhesion is important in many biological processes such as cell migration and proliferation. A better understanding of the cell adhesion process will shed insight into these cellular biological responses as well as cell adhesion-related diseases treatment. However, there is little research which has attempted to investigate the process of cell adhesion and its mechanism. Thus, this paper aims to study the time-dependent adhesion properties of single living chondrocytes using an advanced coupled experimental-numerical approach. Atomic Force Microscopy (AFM) tips will be used to apply lateral forces to detach chondrocytes that are seeded for three different periods. An advanced Finite Element Analysis (FEA) model combining porohyperelastic (PHE) constitutive model and cohesive zone formulation is developed to explore the mechanism of adhesion. The results revealed that the cells can resist normal traction better than tangential traction in the beginning of adhesion. This is when the cell adhesion molecules establish early attachment to the substrates. After that when the cells are spreading, stress fiber bundles generate tangential traction on the substrate to form strong adhesion. Both simulation and experimental results agree well with each other, providing a powerful tool to study the cellular adhesion process.

A Finite Element Model for Mixed Porohyperelasticity with Transport, Swelling, and Growth

The purpose of this manuscript is to establish a unified theory of porohyperelasticity with transport and growth and to demonstrate the capability of this theory using a finite element model developed in MATLAB. We combine the theories of volumetric growth and mixed porohyperelasticity with transport and swelling (MPHETS) to derive a new method that models growth of biological soft tissues. The conservation equations and constitutive equations are developed for both solid-only growth and solid/fluid growth. An axisymmetric finite element framework is introduced for the new theory of growing MPHETS (GMPHETS). To illustrate the capabilities of this model, several example finite element test problems are considered using model geometry and material parameters based on experimental data from a porcine coronary artery. Multiple growth laws are considered, including time-driven, concentration-driven, and stress-driven growth. Time-driven growth is compared against an exact analytical solution to validate the model. For concentration-dependent growth, changing the diffusivity (representing a change in drug) fundamentally changes growth behavior. We further demonstrate that for stress-dependent, solid-only growth of an artery, growth of an MPHETS model results in a more uniform hoop stress than growth in a hyperelastic model for the same amount of growth time using the same growth law. This may have implications in the context of developing residual stresses in soft tissues under intraluminal pressure. To our knowledge, this manuscript provides the first full description of an MPHETS model with growth. The developed computational framework can be used in concert with novel in-vitro and in-vivo experimental approaches to identify the governing growth laws for various soft tissues.

Investigation of the Effects of Extracellular Osmotic Pressure on Morphology and Mechanical Properties of Individual Chondrocyte

It has been demonstrated that most cells of the body respond to osmotic pressure in a systematic manner. The disruption of the collagen network in the early stages of osteoarthritis causes an increase in water content of cartilage which leads to a reduction of pericellular osmolality in chondrocytes distributed within the extracellular environment. It is therefore arguable that an insight into the mechanical properties of chondrocytes under varying osmotic pressure would provide a better understanding of chondrocyte mechanotransduction and potentially contribute to knowledge on cartilage degeneration. In this present study, the chondrocyte cells were exposed to solutions with different osmolality. Changes in their dimensions and mechanical properties were measured over time. Atomic force microscopy (AFM) was used to apply load at various strain-rates and the force-time curves were logged. The thin-layer elastic model was used to extract the elastic stiffness of chondrocytes at different strain-rates and at different solution osmolality. In addition, the porohyperelastic (PHE) model was used to investigate the strain-rate-dependent responses under the loading and osmotic pressure conditions. The results revealed that the hypo-osmotic external environment increased chondrocyte dimensions and reduced Young's modulus of the cells at all strain-rates tested. In contrast, the hyper-osmotic external environment reduced dimensions and increased Young's modulus. Moreover, using the PHE model coupled with inverse FEA simulation, we established that the hydraulic permeability of chondrocytes increased with decreasing extracellular osmolality which is consistent with previous work in the literature. This could be due to a higher intracellular fluid volume fraction with lower osmolality.

Sound transmission in porcine thorax through airway insonification

Many pulmonary injuries and pathologies may lead to structural and functional changes in the lungs resulting in measurable sound transmission changes on the chest surface. Additionally, noninvasive imaging of externally driven mechanical wave motion in the chest (e.g., using magnetic resonance elastography) can provide information about lung structural property changes and, hence, may be of diagnostic value. In the present study, a comprehensive computational simulation (in silico) model was developed to simulate sound wave propagation in the airways, lung, and chest wall under normal and pneumothorax conditions. Experiments were carried out to validate the model. Here, sound waves with frequency content from 50 to 700 Hz were introduced into airways of five porcine subjects via an endotracheal tube, and transmitted waves were measured by scanning laser Doppler vibrometry at the chest wall surface. The computational model predictions of decreased sound transmission with pneumothorax were consistent with experimental measurements. The in silico model can also be used to visualize wave propagation inside and on the chest wall surface for other pulmonary pathologies, which may help in developing and interpreting diagnostic procedures that utilize sound and vibration.

Microscale consolidation analysis of relaxation behavior of single living chondrocytes subjected to varying strain-rates

Besides the elastic stiffness, the relaxation behavior of single living cells is also of interest of various researchers when studying cell mechanics. It is hypothesized that the relaxation response of the cells is governed by both intrinsic viscoelasticity of the solid phase and fluid-solid interactions mechanisms. There are a number of mechanical models have been developed to investigate the relaxation behavior of single cells. However, there is lack of model enable to accurately capture both of the mechanisms. Therefore, in this study, the porohyperelastic (PHE) model, which is an extension of the consolidation theory, combined with inverse Finite Element Analysis (FEA) technique was used at the first time to investigate the relaxation response of living chondrocytes. This model was also utilized to study the dependence of relaxation behavior of the cells on strain-rates. The stress-relaxation experiments under the various strain-rates were conducted with the Atomic Force Microscopy (AFM). The results have demonstrated that the PHE model could effectively capture the stress-relaxation behavior of the living chondrocytes, especially at intermediate to high strain-rates. Although this model gave some errors at lower strain-rates, its performance was acceptable. Therefore, the PHE model is properly a promising model for single cell mechanics studies. Moreover, it has been found that the hydraulic permeability of living chondrocytes reduced with decreasing of strain-rates. It might be due to the intracellular fluid volume fraction and the fluid pore pressure gradients of chondrocytes were higher when higher strain-rates applied.

A computational study of the role of the aortic arch in idiopathic unilateral vocal-fold paralysis

Unilateral vocal-fold paralysis (UVP) occurs when one of the vocal folds becomes paralyzed due to damage to the recurrent laryngeal nerve (RLN). Individuals with UVP experience problems with speaking, swallowing, and breathing. Nearly two-thirds of all cases of UVP is associated with impaired function of the left RLN, which branches from the vagus nerve within the thoracic cavity and loops around the aorta before ascending to the larynx within the neck. We hypothesize that this path predisposes the left RLN to a supraphysiological, biomechanical environment, contributing to onset of UVP. Specifically, this research focuses on the identification of the contribution of the aorta to onset of left-sided UVP. Important to this goal is determining the relative influence of the material properties of the RLN and the aorta in controlling the biomechanical environment of the RLN. Finite element analysis was used to estimate the stress and strain imposed on the left RLN as a function of the material properties and loading conditions. The peak stress and strain in the RLN were quantified as a function of RLN and aortic material properties and aortic blood pressure using Spearman rank correlation coefficients. The material properties of the aortic arch showed the strongest correlation with peak stress [ρ = -0.63, 95% confidence interval (CI), -1.00 to -0.25] and strain (ρ = -0.62, 95% CI, -0.99 to -0.24) in the RLN. Our results suggest an important role for the aorta in controlling the biomechanical environment of the RLN and potentially in the onset of left-sided UVP that is idiopathic.

Multiphasic finite element framework for modeling hydrated mixtures with multiple neutral and charged solutes

Computational tools are often needed to model the complex behavior of biological tissues and cells when they are represented as mixtures of multiple neutral or charged constituents. This study presents the formulation of a finite element modeling framework for describing multiphasic materials in the open-source finite element software febio.1 Multiphasic materials may consist of a charged porous solid matrix, a solvent, and any number of neutral or charged solutes. This formulation proposes novel approaches for addressing several challenges posed by the finite element analysis of such complex materials: The exclusion of solutes from a fraction of the pore space due to steric volume and short-range electrostatic effects is modeled by a solubility factor, whose dependence on solid matrix deformation and solute concentrations may be described by user-defined constitutive relations. These solute exclusion mechanisms combine with long-range electrostatic interactions into a partition coefficient for each solute whose value is dependent upon the evaluation of the electric potential from the electroneutrality condition. It is shown that this electroneutrality condition reduces to a polynomial equation with only one valid root for the electric potential, regardless of the number and valence of charged solutes in the mixture. The equation of charge conservation is enforced as a constraint within the equation of mass balance for each solute, producing a natural boundary condition for solute fluxes that facilitates the prescription of electric current density on a boundary. It is also shown that electrical grounding is necessary to produce numerical stability in analyses where all the boundaries of a multiphasic material are impermeable to ions. Several verification problems are presented that demonstrate the ability of the code to reproduce known or newly derived solutions: (1) the Kedem-Katchalsky model for osmotic loading of a cell; (2) Donnan osmotic swelling of a charged hydrated tissue; and (3) current flow in an electrolyte. Furthermore, the code is used to generate novel theoretical predictions of known experimental findings in biological tissues: (1) current-generated stress in articular cartilage and (2) the influence of salt cation charge number on the cartilage creep response. This generalized finite element framework for multiphasic materials makes it possible to model the mechanoelectrochemical behavior of biological tissues and cells and sets the stage for the future analysis of reactive mixtures to account for growth and remodeling.

What have we learned from finite element model studies of lumbar intervertebral discs in the past four decades?

Finite element analysis is a powerful tool routinely used to study complex biological systems. For the last four decades, the lumbar intervertebral disc has been the focus of many such investigations. To understand the disc functional biomechanics, a precise knowledge of the disc mechanical, structural and biochemical environments at the microscopic and macroscopic levels is essential. In response to this need, finite element model studies have proven themselves as reliable and robust tools when combined with in vitro and in vivo measurements. This paper aims to review and discuss some salient findings of reported finite element simulations of lumbar intervertebral discs with special focus on their relevance and implications in disc functional biomechanics. Towards this goal, the earlier investigations are presented, discussed and summarized separately in three distinct groups of elastic, multi-phasic transient and transport model studies. The disc overall response as well as the relative role of its constituents are markedly influenced by loading rate, magnitude, combinations/preloads and posture. The nucleus fluid content and pressurizing capacity affect the disc compliance, annulus strains and failure sites/modes. Biodynamics of the disc is affected by not only the excitation characteristics but also preloads, existing mass and nucleus condition. The role of fluid pressurization and collagen fiber stiffening diminish with time during diurnal loading. The endplates permeability influences the time-dependent response of the disc in both loaded and unloaded recovery phases. The transport of solutes is substantially influenced by the disc size, tissue diffusivity and endplates permeability.

A review of the combination of experimental measurements and fibril-reinforced modeling for investigation of articular cartilage and chondrocyte response to loading

The function of articular cartilage depends on its structure and composition, sensitively impaired in disease (e.g. osteoarthritis, OA). Responses of chondrocytes to tissue loading are modulated by the structure. Altered cell responses as an effect of OA may regulate cartilage mechanotransduction and cell biosynthesis. To be able to evaluate cell responses and factors affecting the onset and progression of OA, local tissue and cell stresses and strains in cartilage need to be characterized. This is extremely challenging with the presently available experimental techniques and therefore computational modeling is required. Modern models of articular cartilage are inhomogeneous and anisotropic, and they include many aspects of the real tissue structure and composition. In this paper, we provide an overview of the computational applications that have been developed for modeling the mechanics of articular cartilage at the tissue and cellular level. We concentrate on the use of fibril-reinforced models of cartilage. Furthermore, we introduce practical considerations for modeling applications, including also experimental tests that can be combined with the modeling approach. At the end, we discuss the prospects for patient-specific models when aiming to use finite element modeling analysis and evaluation of articular cartilage function, cellular responses, failure points, OA progression, and rehabilitation.

Computational aspects in mechanical modeling of the articular cartilage tissue

This review focuses on the modeling of articular cartilage (at the tissue level), chondrocyte mechanobiology (at the cell level) and a combination of both in a multiscale computation scheme. The primary objective is to evaluate the advantages and disadvantages of conventional models implemented to study the mechanics of the articular cartilage tissue and chondrocytes. From monophasic material models as the simplest form to more complicated multiscale theories, these approaches have been frequently used to model articular cartilage and have contributed significantly to modeling joint mechanics, addressing and resolving numerous issues regarding cartilage mechanics and function. It should be noted that attentiveness is important when using different modeling approaches, as the choice of the model limits the applications available. In this review, we discuss the conventional models applicable to some of the mechanical aspects of articular cartilage such as lubrication, swelling pressure and chondrocyte mechanics and address some of the issues associated with the current modeling approaches. We then suggest future pathways for a more realistic modeling strategy as applied for the simulation of the mechanics of the cartilage tissue using multiscale and parallelized finite element method.

Finite element implementation of mechanochemical phenomena in neutral deformable porous media under finite deformation

Biological soft tissues and cells may be subjected to mechanical as well as chemical (osmotic) loading under their natural physiological environment or various experimental conditions. The interaction of mechanical and chemical effects may be very significant under some of these conditions, yet the highly nonlinear nature of the set of governing equations describing these mechanisms poses a challenge for the modeling of such phenomena. This study formulated and implemented a finite element algorithm for analyzing mechanochemical events in neutral deformable porous media under finite deformation. The algorithm employed the framework of mixture theory to model the porous permeable solid matrix and interstitial fluid, where the fluid consists of a mixture of solvent and solute. A special emphasis was placed on solute-solid matrix interactions, such as solute exclusion from a fraction of the matrix pore space (solubility) and frictional momentum exchange that produces solute hindrance and pumping under certain dynamic loading conditions. The finite element formulation implemented full coupling of mechanical and chemical effects, providing a framework where material properties and response functions may depend on solid matrix strain as well as solute concentration. The implementation was validated using selected canonical problems for which analytical or alternative numerical solutions exist. This finite element code includes a number of unique features that enhance the modeling of mechanochemical phenomena in biological tissues. The code is available in the public domain, open source finite element program FEBio (http:∕∕mrl.sci.utah.edu∕software).

Solute transport across a contact interface in deformable porous media

A finite element formulation of neutral solute transport across a contact interface between deformable porous media is implemented and validated against analytical solutions. By reducing the integral statements of external virtual work on the two contacting surfaces into a single contact integral, the algorithm automatically enforces continuity of solute molar flux across the contact interface, whereas continuity of the effective solute concentration (a measure of the solute mechano-chemical potential) is achieved using a penalty method. This novel formulation facilitates the analysis of problems in biomechanics where the transport of metabolites across contact interfaces of deformable tissues may be of interest. This contact algorithm is the first to address solute transport across deformable interfaces, and is made available in the public domain, open-source finite element code FEBio (http://www.febio.org).

Biomechanical Multiphase Approaches in Soft Tissues

Presence of fluids drives mechanical response in biological, soft hydrated tissues. A multiphase approach has been developed to allow for a fully-comprehensive, physically based view of their history-dependent response in static and dynamics. The related formulation considers both geometric and material non-linearities. In the present work, the non-linear aspects related to fluid flows are investigated. Low speed flows of liquid phase depend on the permeability of the deforming solid matrix; permeability is naturally related to void ratio, but further dependences on other variables can be explicitly added. Two different biomechanical problems are considered and the results are presented here. In the first case, a human lumbar intervertebral segment is investigated in healthy and degenerate condition: higher permeability and lower mechanical properties of disc affect the capability to withstand an imposed constant compressive load in time. The permeability is related only to void ratio and is not affected by the overpressure that dissipates with time. The second case refers to the trabecular meshwork in the eye; this tissue has a rule in the regulation of outflow of the aqueous humour from the eye: a variation in permeability determines obviously a variation of internal ocular pressure to guarantee the assumed constant outflow, and on the other side, pressure changes affect permeability distribution and thus final pressure changes. This effect is particularly significant with glaucoma, and thus the increment of intraocular pressure can be related to a progressive reduction in tissue permeability. The modelling of this effect can be improved by introducing a proper permeability-pressure relationship in addition to the permeability-void ratio dependency.

Comparison of four methods to simulate swelling in poroelastic finite element models of intervertebral discs

Osmotic phenomena influence the intervertebral disc biomechanics. Their simulation is challenging and can be undertaken at different levels of complexity. Four distinct approaches to simulate the osmotic behaviour of the intervertebral disc (a fixed boundary pore pressure model, a fixed osmotic pressure gradient model in the whole disc or only in the nucleus pulposus, and a swelling model with strain-dependent osmotic pressure) were analysed. Predictions were compared using a 3D poroelastic finite element model of a L4-L5 spinal unit under three different loading conditions: free swelling for 8 h and two daily loading cycles: (i) 200 N compression for 8 h followed by 500 N compression for 16 h; (ii) 500 N for 8 h followed by 1000 N for 16 h. Overall, all swelling models calculated comparable results, with differences decreasing under greater loads. Results predicted with the fixed boundary pore pressure and the fixed osmotic pressure in the whole disc models were nearly identical. The boundary pore pressure model, however, cannot simulate differential osmotic pressures in disc regions. The swelling model offered the best potential to provide more accurate results, conditional upon availability of reliable values for the required coefficients and material properties. Possible fields of application include mechanobiology investigations and crack opening and propagation. However, the other approaches are a good compromise between the ease of implementation and the reliability of results, especially when considering higher loads or when the focus is on global results such as spinal kinematics.

DYNAMICAL FINITE ELEMENT MODELING OF SOFT TISSUES AS CHEMOELECTRIC POROUS MEDIA

An implicit mixed finite element formulation of hydrated soft biological tissues, based on the Simon model, is presented that incorporates the coupling of solid, fluid, and ion phases as well as the viscoelasticity of soft tissue in the dynamical process. The tissues are modeled as a multi-field viscoelastic body subject to finite deformation. In addition to a three-field (u-w-p) modeling of the porous matrix, the study also includes an ion phase for the ionic solution. After presenting the formulation, an efficient staggered solution scheme is presented: within each time step, the ion charge equation is solved first to give the distribution of the charge concentration, the charge induced osmotic water pressure is then employed in solving the u-w-p equations. The resulting u field becomes a forcing term to the solution of the ion charge concentration equations for iteration. This methodology and codes developed for the study have been verified with one-dimensional (1D) analytical solutions. A 2D chemical electric swelling model illustrates the important role of viscoelasticity. A brain tissue impact example demonstrates the potential application of the model.

Mathematical Modeling of the Circulation in the Liver Lobule

In this paper, we develop a mathematical model of blood circulation in the liver lobule. We aim to find the pressure and flux distributions within a liver lobule. We also investigate the effects of changes in pressure that occur following a resection of part of the liver, which often leads to high pressure in the portal vein. The liver can be divided into functional units called lobules. Each lobule has a hexagonal cross-section, and we assume that its longitudinal extent is large compared with its width. We consider an infinite lattice of identical lobules and study the two-dimensional flow in the hexagonal cross-sections. We model the sinusoidal space as a porous medium, with blood entering from the portal tracts (located at each of the vertices of the cross-section of the lobule) and exiting via the centrilobular vein (located in the center of the cross-section). We first develop and solve an idealized mathematical model, treating the porous medium as rigid and isotropic and blood as a Newtonian fluid. The pressure drop across the lobule and the flux of blood through the lobule are proportional to one another. In spite of its simplicity, the model gives insight into the real pressure and velocity distribution in the lobule. We then consider three modifications of the model that are designed to make it more realistic. In the first modification, we account for the fact that the sinusoids tend to be preferentially aligned in the direction of the centrilobular vein by considering an anisotropic porous medium. In the second, we account more accurately for the true behavior of the blood by using a shear-thinning model. We show that both these modifications have a small quantitative effect on the behavior but no qualitative effect. The motivation for the final modification is to understand what happens either after a partial resection of the liver or after an implantation of a liver of small size. In these cases, the pressure is observed to rise significantly, which could cause deformation of the tissue. We show that including the effects of tissue compliance in the model means that the total blood flow increases more than linearly as the pressure rises.

Refinement of elastic, poroelastic, and osmotic tissue properties of intervertebral disks to analyze behavior in compression

Intervertebral disks support compressive forces because of their elastic stiffness as well as the fluid pressures resulting from poroelasticity and the osmotic (swelling) effects. Analytical methods can quantify the relative contributions, but only if correct material properties are used. To identify appropriate tissue properties, an experimental study and finite element analytical simulation of poroelastic and osmotic behavior of intervertebral disks were combined to refine published values of disk and endplate properties to optimize model fit to experimental data. Experimentally, nine human intervertebral disks with adjacent hemi-vertebrae were immersed sequentially in saline baths having concentrations of 0.015, 0.15, and 1.5 M and the loss of compressive force at constant height (force relaxation) was recorded over several hours after equilibration to a 300-N compressive force. Amplitude and time constant terms in exponential force-time curve-fits for experimental and finite element analytical simulations were compared. These experiments and finite element analyses provided data dependent on poroelastic and osmotic properties of the disk tissues. The sensitivities of the model to alterations in tissue material properties were used to obtain refined values of five key material parameters. The relaxation of the force in the three bath concentrations was exponential in form, expressed as mean compressive force loss of 48.7, 55.0, and 140 N, respectively, with time constants of 1.73, 2.78, and 3.40 h. This behavior was analytically well represented by a model having poroelastic and osmotic tissue properties with published tissue properties adjusted by multiplying factors between 0.55 and 2.6. Force relaxation and time constants from the analytical simulations were most sensitive to values of fixed charge density and endplate porosity.

On the thermodynamical admissibility of the triphasic theory of charged hydrated tissues

The triphasic theory on soft charged hydrated tissues (Lai, W. M., Hou, J. S., and Mow, V. C., 1991, "A Triphasic Theory for the Swelling and Deformation Behaviors of Articular Cartilage," ASME J. Biomech. Eng., 113, pp. 245-258) attributes the swelling propensity of articular cartilage to three different mechanisms: Donnan osmosis, excluded volume effect, and chemical expansion stress. The aim of this study is to evaluate the thermodynamic plausibility of the triphasic theory. The free energy of a sample of articular cartilage subjected to a closed cycle of mechanical and chemical loading is calculated using the triphasic theory. It is shown that the chemical expansion stress term induces an unphysiological generation of free energy during each closed cycle of loading and unloading. As the cycle of loading and unloading can be repeated an indefinite number of times, any amount of free energy can be drawn from a sample of articular cartilage, if the triphasic theory were true. The formulation for the chemical expansion stress as used in the triphasic theory conflicts with the second law of thermodynamics.

Assessment of common hyperelastic constitutive equations for describing normal and osteoarthritic articular cartilage

With the aim of providing information for modelling joint and limb systems, widely available constitutive hyperelastic laws are evaluated in this paper for their ability to predict the mechanical responses of normal and osteoarthritic articular cartilage. Load—displacement data from mechanical indentation were obtained for normal and osteoarthritic cartilage at 0.1 s−1 and 0.025 s−1 and converted to the stress—stretch ratio. The data were then fitted to the ArrudA—Boyce, Mooney—Rivlin, neo-Hookean, Ogden, polynomial, and Yeoh hyperelastic laws in the MATLAB environment. Although each of the hyperelastic laws performed satisfactorily at the higher rate of loading, their ability to fit experimental data at the lower loading rate varied considerably. For the preferred models, coefficients were provided for stiff, soft, and average tissues to represent normal and degraded tissue at high and low loading rates. The present authors recommend the use of the Mooney—Rivlin or the Yeoh models for describing both normal and degraded articular cartilage, with the Mooney—Rivlin model providing the best compromise between accuracy and required computational power.

Modeling proliferative tissue growth: a general approach and an avian case study

During development, tissues often undergo rapid physical expansion due to cell proliferation. Continuous and discrete models of one- and two-dimensional tissue growth are developed and applied to observational data of the developing avian gut, where the gut tissue cells undergo dramatic proliferation. The discrete cellular automata model provides results at the level of individual cells that reflect a realistic stochasticity and nonuniformity expected in cellular systems. Averaging the discrete results predicts population-level properties of the system, which match those of the continuous model. This dual approach provides an understanding of the interaction between the individual-level and population-level aspects of a developmental growth process. Both models are applied to a case study involving the developing intestinal tract of a quail embryo. A nonuniform growth model accurately predicts the positions of measurable biological landmarks within the growing tissue. Furthermore, the discrete model provides a framework for modeling the interactions between growing tissues and other biological mechanisms, such as cell motility and proliferation on an expanding tissue.

Poromechanics of compressible charged porous media using the theory of mixtures

Osmotic, electrostatic, and/or hydrational swellings are essential mechanisms in the deformation behavior of porous media, such as biological tissues, synthetic hydrogels, and clay-rich rocks. Present theories are restricted to incompressible constituents. This assumption typically fails for bone, in which electrokinetic effects are closely coupled to deformation. An electrochemomechanical formulation of quasistatic finite deformation of compressible charged porous media is derived from the theory of mixtures. The model consists of a compressible charged porous solid saturated with a compressible ionic solution. Four constituents following different kinematic paths are identified: a charged solid and three streaming constituents carrying either a positive, negative, or no electrical charge, which are the cations, anions, and fluid, respectively. The finite deformation model is reduced to infinitesimal theory. In the limiting case without ionic effects, the presented model is consistent with Blot's theory. Viscous drag compression is computed under closed circuit and open circuit conditions. Viscous drag compression is shown to be independent of the storage modulus. A compressible version of the electrochemomechanical theory is formulated. Using material parameter values for bone, the theory predicts a substantial influence of density changes on a viscous drag compression simulation. In the context of quasistatic deformations, conflicts between poromechanics and mixture theory are only semantic in nature.

Toward a model for local drug delivery in abdominal aortic aneurysms

The formation of an abdominal aortic aneurysm (AAA) may eventually result in rupture, an event associated with a 50% mortality rate. This work represents a first step toward improving current stress estimation techniques and local transport simulations in AAA. Toward this aim, a computational parametric study was performed on an axisymmetric cylindrical FEM of a 5 cm AAA with a 1.5 cm thick intraluminal thrombus (ILT). Both the AAA wall and ILT were modeled as porohyperelastic PHE materials using estimated values of AAA wall and ILT permeability. While no values for AAA wall permeability could be found in the literature, the value of ILT permeability was taken from a previous investigation by Adolph et al.(7) Peak stresses, fluid velocities, and local pore pressure values within the ILT and wall were recorded and analyzed as a function of the cardiac cycle. While peak wall stress values for the PHE models did not largely differ from corresponding solid finite element simulations (186.2 N/cm(2) vs. 186.5 N/cm(2)), the stress in the abluminal region of the ILT increased by 17.4% (7.7 N/cm(2) vs. 6.5 N/cm(2)). Pore pressure values were relatively constant through the ILT while there were significant pore pressure gradients present in the AAA wall. The magnitude of fluid velocities varied in magnitude and direction throughout the cardiac cycle with large fluctuations occurring on the luminal surface. The combination of the patient-specific PHE AAA FEMs with mass transport simulations will result in spatially and time-varying concentration distributions within AAA, which may benefit future pharmaceutical treatments of AAA.

Dependence of mechanical behavior of the murine tail disc on regional material properties: a parametric finite element study

In vivo rodent tail models are becoming more widely used for exploring the role of mechanical loading on the initiation and progression of intervertebral disc degeneration. Historically, finite element models (FEMs) have been useful for predicting disc mechanics in humans. However, differences in geometry and tissue properties may limit the predictive utility of these models for rodent discs. Clearly, models that are specific for rodent tail discs and accurately simulate the disc's transient mechanical behavior would serve as important tools for clarifying disc mechanics in these animal models. An FEM was developed based on the structure, geometry, and scale of the mouse tail disc. Importantly, two sources of time-dependent mechanical behavior were incorporated: viscoelasticity of the matrix, and fluid permeation. In addition, a novel strain-dependent swelling pressure was implemented through the introduction of a dilatational stress in nuclear elements. The model was then validated against data from quasi-static tension-compression and compressive creep experiments performed previously using mouse tail discs. Finally, sensitivity analyses were performed in which material parameters of each disc subregion were individually varied. During disc compression, matrix consolidation was observed to occur preferentially at the periphery of the nucleus pulposus. Sensitivity analyses revealed that disc mechanics was greatly influenced by changes in nucleus pulposus material properties, but rather insensitive to variations in any of the endplate properties. Moreover, three key features of the model-nuclear swelling pressure, lamellar collagen viscoelasticity, and interstitial fluid permeation-were found to be critical for accurate simulation of disc mechanics. In particular, collagen viscoelasticity dominated the transient behavior of the disc during the initial 2200 s of creep loading, while fluid permeation governed disc deformation thereafter. The FEM developed in this study exhibited excellent agreement with transient creep behavior of intact mouse tail motion segments. Notably, the model was able to produce spatial variations in nucleus pulposus matrix consolidation that are consistent with previous observations in nuclear cell morphology made in mouse discs using confocal microscopy. Results of this study emphasize the need for including nucleus swelling pressure, collagen viscoelasticity, and fluid permeation when simulating transient changes in matrix and fluid stress/strain. Sensitivity analyses suggest that further characterization of nucleus pulposus material properties should be pursued, due to its significance in steady-state and transient disc mechanical response.

Dynamic finite element modeling of poroviscoelastic soft tissue

Clinical evidences relative to biomechanical factors have demonstrated their important contribution to the behaviour of soft tissues. Finite element (FE) analysis is used to study the mechanical behaviour of soft tissue because it can provide numerical solutions to problems that are intractable to analytic solutions. This study focuses on the development of a FE model of a poroelastic biological tissue, which incorporates the viscoelastic material behaviour, finite deformation and inertial effect. The FE formulation is based on the weak form derived from the governing equation, and Newmark-beta method as well as Newton's method is incorporated into the implicit non-linear solutions. One-dimensional analytical solutions were used to verify the theoretical formulation and the numerical implementation of the proposed model. This study was further extended to analyze two-dimensional biomechanical models and the results clearly demonstrate the importance of including finite deformation, viscoelasticity and inertial effects.

A comparison between mechano-electrochemical and biphasic swelling theories for soft hydrated tissues

Biological tissues like intervertebral discs and articular cartilage primarily consist of interstitial fluid, collagen fibrils and negatively charged proteoglycans. Due to the fixed charges of the proteoglycans, the total ion concentration inside the tissue is higher than in the surrounding synovial fluid (cation concentration is higher and the anion concentration is lower). This excess of ion particles leads to an osmotic pressure difference, which causes swelling of the tissue. In the last decade several mechano-electrochemical models, which include this mechanism, have been developed. As these models are complex and computationally expensive, it is only possible to analyze geometrically relatively small problems. Furthermore, there is still no commercial finite element tool that includes such a mechano-electrochemical theory. Lanir (Biorheology, 24, pp. 173-187, 1987) hypothesized that electrolyte flux in articular cartilage can be neglected in mechanical studies. Lanir's hypothesis implies that the swelling behavior of cartilage is only determined by deformation of the solid and by fluid flow. Hence, the response could be described by adding a deformation-dependent pressure term to the standard biphasic equations. Based on this theory we developed a biphasic swelling model. The goal of the study was to test Lanir's hypothesis for a range of material properties. We compared the deformation behavior predicted by the biphasic swelling model and a full mechano-electrochemical model for confined compression and 1D swelling. It was shown that, depending on the material properties, the biphasic swelling model behaves largely the same as the mechano-electrochemical model, with regard to stresses and strains in the tissue following either mechanical or chemical perturbations. Hence, the biphasic swelling model could be an alternative for the more complex mechano-electrochemical model, in those cases where the ion flux itself is not the subject of the study. We propose thumbrules to estimate the correlation between the two models for specific problems.

Cell mechanics and mechanobiology in the intervertebral disc

STUDY DESIGN

A review is presented on current knowledge of the micromechanical factors in the intervertebral disc, their role in modifying cell biology, and changes with degeneration.

OBJECTIVES

To identify current knowledge, knowledge gaps, and areas for future research in micromechanics of the intervertebral disc.

SUMMARY OF BACKGROUND DATA

Mechanical factors play important roles in the initiation and progression of intervertebral disc degeneration. Evidence suggests that substantial biologic remodeling occurs in the intervertebral disc in response to mechanical stimuli that may play a critical role in determining the fate of a degenerating intervertebral disc. Information is needed on the precise mechanical stimuli that these cells experience and the mechanisms that govern their responses.

METHODS

A review is presented of cell morphology, cell mechanics, and the internal strains and other mechanical factors predicted to occur at the cell level. A review of intervertebral disc cell responses to well-controlled physical stimuli is also presented with a focus on in vitro studies of explants and isolated cells.

RESULTS

Important differences in cell morphology, mechanics, micromechanical factors, and mechanobiology are noted to occur between cells of the nucleus pulposus and anulus fibrosus. Changes in these features with degeneration are critically understudied, particularly degeneration-associated changes in cell morphology, cell mechanics, and altered physiology with mechanical loading.

CONCLUSIONS

Information on the mechanisms that govern cell responses to mechanical stimuli in the intervertebral disc are just emerging. Studies must address determination of the factors that control micromechanical stimuli, but also mechanisms by which mechanics may interact with genetic factors to regulate expression and remodeling of extracellular matrix molecules, cytokines and mediators of pain and inflammation in degenerating tissue.

A finite element formulation and program to study transient swelling and load-carriage in healthy and degenerate articular cartilage

The theory of poroelasticity is extended to include physico-chemical swelling and used to predict the transient responses of normal and degenerate articular cartilage to both chemical and mechanical loading; with emphasis on isolating the influence of the major parameters which govern its deformation. Using a new hybrid element, our mathematical relationships were implemented in a purpose-built poroelastic finite element analysis algorithm (u-pi-c program) which was used to resolve the nature of the coupling between the mechanical and chemical responses of cartilage when subjected to ionic transport across its membranous skeleton. Our results demonstrate that one of the roles of the strain-dependent matrix permeability is to limit the rate of transmission of stresses from the fluid to the collagen-proteoglycan solid skeleton in the incipient stages of loading, and that the major contribution of the swelling pressure is that of preventing any excessive deformation of the matrix.

Fluid flow and convective transport of solutes within the intervertebral disc

Previous experimental and analytical studies of solute transport in the intervertebral disc have demonstrated that for small molecules diffusive transport alone fulfils the nutritional needs of disc cells. It has been often suggested that fluid flow into and within the disc may enhance the transport of larger molecules. The goal of the study was to predict the influence of load-induced interstitial fluid flow on mass transport in the intervertebral disc. An iterative procedure was used to predict the convective transport of physiologically relevant molecules within the disc. An axisymmetric, poroelastic finite-element structural model of the disc was developed. The diurnal loading was divided into discrete time steps. At each time step, the fluid flow within the disc due to compression or swelling was calculated. A sequentially coupled diffusion/convection model was then employed to calculate solute transport, with a constant concentration of solute being provided at the vascularised endplates and outer annulus. Loading was simulated for a complete diurnal cycle, and the relative convective and diffusive transport was compared for solutes with molecular weights ranging from 400 Da to 40 kDa. Consistent with previous studies, fluid flow did not enhance the transport of low-weight solutes. During swelling, interstitial fluid flow increased the unidirectional penetration of large solutes by approximately 100%. Due to the bi-directional temporal nature of disc loading, however, the net effect of convective transport over a full diurnal cycle was more limited (30% increase). Further study is required to determine the significance of large solutes and the timing of their delivery for disc physiology.

A simulation model for analysing brain structure deformations

Recent developments of medical software applications--from the simulation to the planning of surgical operations--have revealed the need for modelling human tissues and organs, not only from a geometric point of view but also from a physical one, i.e. soft tissues, rigid body, viscoelasticity, etc. This has given rise to the term 'deformable objects', which refers to objects with a morphology, a physical and a mechanical behaviour of their own and that reflects their natural properties. In this paper, we propose a model, based upon physical laws, suitable for the realistic manipulation of geometric reconstructions of volumetric data taken from MR and CT scans. In particular, a physically based model of the brain is presented that is able to simulate the evolution of different nature pathological intra-cranial phenomena such as haemorrhages, neoplasm, haematoma, etc and to describe the consequences that are caused by their volume expansions and the influences they have on the anatomical and neuro-functional structures of the brain.

Influence of fixed charge density magnitude and distribution on the intervertebral disc: applications of a poroelastic and chemical electric (PEACE) model

A 3-dimensional formulation for a poroelastic and chemical electric (PEACE) model is presented and applied to an intervertebral disc slice in a 1-dimensional validation problem and a 2-dimensional plane stress problem. The model was used to investigate the influence of fixed charge density magnitude and distribution on this slice of disc material. Results indicated that the mechanical, chemical, and electrical behaviors were all strongly influenced by the amount as well as the distribution of fixed charges in the matrix. Without any other changes in material properties, alterations in the fixed charge density (proteoglycan content) from a healthy to a degenerated distribution will cause an increase in solid matrix stresses and can affect whether the tissue imbibes or exudes fluid under different loading conditions. Disc tissue with a degenerated fixed charge density distribution exhibited greater solid matrix stresses and decreased streaming potential, all of which have implications for disc nutrition, disc biomechanics, and tissue remodeling. It was also seen that application of an electrical potential across the disc can induce fluid transport.

Simulating the swelling and deformation behaviour in soft tissues using a convective thermal analogy

BACKGROUND

It is generally accepted that cartilage adaptation and degeneration are mechanically mediated. Investigating the swelling behaviour of cartilage is important because the stress and strain state of cartilage is associated with the swelling and deformation behaviour. It is well accepted that the swelling of soft tissues is associated with mechanical, chemical, and electrical events.

METHOD

The purpose of the present study was to implement the triphasic theory into a commercial finite element tool (ABAQUS) to solve practical problems in cartilage mechanics. Because of the mathematical identity between thermal and mass diffusion processes, the triphasic model was transferred into a convective thermal diffusion process in the commercial finite element software. The problem was solved using an iterative procedure.

RESULTS

The proposed approach was validated using the one-dimensional numerical solutions and the experimental results of confined compression of articular cartilage described in the literature. The time-history of the force response of a cartilage specimen in confined compression, which was subjected to swelling caused by a sudden change of saline concentration, was predicted using the proposed approach and compared with the published experimental data.

CONCLUSION

The advantage of the proposed thermal analogy technique over previous studies is that it accounts for the convective diffusion of ion concentrations and the Donnan osmotic pressure in the interstitial fluid.

Multiphase mechanics of capsule formation in tumors

The presence of a capsule around a tumor is known to be correlated with benign status, and the absence of a capsule often has negative implications for patient prognosis. A mechanical description is presented of the growth of a tumor and the resulting deformations of surrounding normal tissue. A mathematical model of the mechanics is analyzed using physical parameters measured in vivo and in vitro. The model has only three dimensionless parameters, and its results are very robust with respect to parameter variation. We show that the presence of contractility in the surrounding tissue, corresponding to a host defense, can make an existing capsule denser and constrain the tumor better, but cannot be responsible for the observed pressure gradients in encapsulated tumors. Some implications for treatment are discussed.

Elastic energy storage in human articular cartilage: estimation of the elastic modulus for type II collagen and changes associated with osteoarthritis

The viscoelastic mechanical properties of normal and osteoarthritic articular were analyzed based on data reported by Kempson [in: Adult Articular Cartilage (1973)] and Silver et al. (Connect. Tissue Res., 2001b). Results of the analysis of tensile elastic stress-strain curves suggest that the elastic modulus of cartilage from the superficial zone is approximately 7.0 GPa parallel and 2.21 GPa perpendicular to the cleavage line pattern. Collagen fibril lengths in the superficial zone were found to be approximately 1265 microm parallel and 668 microm perpendicular to the cleavage line direction. The values for the elastic modulus and fibril lengths decreased with increased extent of osteoarthritis. The elastic modulus of type II collagen parallel to the cleavage line pattern in the superficial zone approaches that of type I collagen in tendon, suggesting that elastic energy storage occurs in the superficial zone due to the tensile pre-tension that exists in this region. Decreases in the elastic modulus associated with osteoarthritis reflect decreased ability of cartilage to store elastic energy, which leads to cartilage fibrillation and fissure formation. We hypothesize that under normal physiological conditions, collagen fibrils in cartilage function to store elastic energy associated with weight bearing and locomotion. Enzymatic cleavage of cartilage proteoglycans and collagen observed in osteoarthritis may lead to fibrillation and fissure formation as a result of impaired energy storage capability of cartilage.

Coupled macroscopic and microscopic scale modeling of fibrillar tissues and tissue equivalents

Collagen mechanics are crucial to the function and dysfunction of many tissues, including blood vessels and articular cartilage, and bioartificial tissues. Previous attempts to develop computer simulations of collagenous tissue based on macroscopic property descriptions have often been limited in application by the simplicity of the model; simulations based on microscopic descriptions, in contrast, have numerical limitations imposed by the size of the mathematical problem. We present a method that combines the tractability of the macroscopic approach with the flexibility of the microstructural approach. The macroscopic domain is divided into finite elements (as in standard FEM). Each element contains a microscopic scale network. Instead of a stress constitutive equation; the macroscopic problem is distributed over the microscopic scale network and solved in each element to satisfy the weak formulation of Cauchy's stress continuity equation over the macroscopic domain. The combined method scales by order 1.1 as the overall number of degrees of freedom is increased, allowing it to handle larger problems than a direct microstructural approach. Model predictions agree qualitatively with tensile tests on isotropic and aligned reconstituted type I collagen gels.

A Porous Media Approach to Finite Deformation Behaviour in Soft Tissues

The present work presents a porous medium formulation for the biomechanical analysis of soft tissues. An updated Lagrangian approach is developed to study the coupled effects of low speed flows of fluid phases, in partially or fully saturated conditions, and the finite deformation occurring in the solid matrix. The procedure developed allows both for the evaluation of coupled geometric and material non-linearities. The main theoretical and computational aspects of this multiphase formulation are discussed. The finite element method is used for the numerical solution of the resulting coupled system of equations. A reference case is reported with regard to healthy and degenerative phases of intervertebral segment. Results reported allow for a detailed interpretation of the formulation reliability, also by comparison with existing experimental data. In particular, the role played by the fluid on the load carrying mechanism is pointed out, thus stressing the importance of a multiphase approach to the overall behaviour of the spinal motion segment in time.

Thermal and biomechanical parameters of porcine cornea

PURPOSE

New methods in refractive surgery require a considerable understanding of the material "cornea" and are often studied by theoretical modeling in order to gain insight into the procedure and an optimized approach to the technique. The quality of these models is highly dependent on the preciseness of its input parameters. Porcine cornea often is used as a model in preclinical studies because of its similarity to man and its availability.

METHODS

The important physical parameters for biomechanical deformation, heat conduction, and collagen denaturation kinetics have been determined for porcine cornea. Experimental methods include densitometry, calorimetry, turbidimetry, tensile tests, stress relaxation, and hydrothermal isometric tension measurements.

RESULTS

The density of porcine cornea was measured as p = 1062+/-5 kg/m3, the heat capacity gave c = 3.74+/-0.05 J/gK. The stress-strain relation for corneal strips is represented by a third order approximation where the secant modulus yields about Esec approximately equal to 0.4 MPa for small strains less than 2%. The normalized stress relaxation is described by an exponential fit over time. The denaturation process of cornea is characterized by specific temperatures which can be related to the change of the mechanical properties. Denaturation kinetics are described according to the model of Arrhenius yielding the activation energy deltaEa = 106 kJ/mol and the phase transition entropy deltaS = 39 J/(mol x K).

CONCLUSIONS

The established set of parameters characterizes the porcine cornea in a reliable way that creates a basis for corneal models. It furthermore gives direct hints of how to treat cornea in certain refractive techniques.

Effect of loading rate and hydration on the mechanical properties of the disc

STUDY DESIGN

The mechanical response of bovine intervertebral discs to axial compression at different loading rates and hydration levels was quantified.

OBJECTIVES

To quantify the effects of hydration and loading rate on the mechanical response of the intervertebral disc to compressive axial load.

SUMMARY OF BACKGROUND DATA

The disc is known to be viscoelastic, but there are few experimental data showing the effect of loading rate and hydration on its response to compression.

METHODS

Hydration level reduced by creep-loading from a fully hydrated starting point. Four groups were tested: Group A: fully hydrated (n = 5), six loading rates, from 0.3 kPa/sec to 30 MPa/sec; Group B: after 30 minutes of creep (n = 4); and Group C: after 2 hours of creep (n = 4) under a static load of 1 MPa, loading rates 3 MPa/sec, 30 kPa/sec, and 0.3 kPa/sec; Group D: at 5-minute intervals, during an 8-hour period of creep (n = 3) under a static load of 1 MPa, loading rate 3 MPa/sec. Data normalized by disc area and height: nominal stress, strain, and modulus calculated.

RESULTS

Group A: Modulus increased with load and rate of loading, with significant differences among the lower three loading rates. The highest three loading rates were significantly different from the lower rates, but not from each other. Group B: At the two higher loading rates, modulus was greater than in group A. At the lowest loading rate the modulus was similar to that in Group A. Group C: At the highest loading rate, the modulus was less than that of Groups A and B. At the lower two loading rates, the modulus was similar to that in Group A. Group D: The modulus increased in the first 30 minutes and decreased in the interval from 60 to 480 minutes.

CONCLUSIONS

Intervertebral disc compressive mechanical properties are significantly dependent on loading rate and hydration.

Material characterization of the pig kidney in relation with the biomechanical analysis of renal trauma

The objective of this study was an investigation of the material properties of the fresh pig kidney and parametric characterization of its elastic and inelastic material behavior. The material investigation included density measurements, uniaxial as well as three-dimensional compression tests, tensile tests. and shear tests on the samples extracted from the fresh pig kidney. For comparison, density measurements on a number of soft synthetic materials were also performed. Compression tests on the radial and the tangential specimens from the cortex tissue were performed at various loading rates. Three-axial compression tests were performed on the cortex tissues placed in a compression chamber. Shear tests were performed by punching a cylinder into a slice of the cortex. Tensile tests were carried out on the outer capsule. For characterization of the material behavior, a non-linear theoretical simulation based on a two parameter Blatz model was used. For characterization of the time-dependent behavior of the pig kidney cortex, a four-parameter linear viscoelastic model was employed. From the present experimental and theoretical studies, a number of conclusions were derived: (1) The general behavior of the pig kidney cortex samples under compression showed the general non-linear features typical of the soft tissues; the stress strain diagram was composed of a very flat part at very low stress level to about 30% relative deformation which was followed by a steeply rising stiffening leading to the radial rupture of samples marked by a maximum nominal rupture strain of about 50%. (2) The uniaxial compression tests on the radial and the tangential samples from the cortex tissue showed an increase of the rupture stress with the increase in the loading rate, but a decrease in the related rupture strain. (3) The long-term uniaxial compression tests on the cortex specimens under sustained constant load showed an instantaneous deformation followed by a creep response which eventually approached an asymptote. (4) Simulation of the non-linear material behavior of the cortex tissue under uniaxial compression by the Blatz model gave two pairs of material parameters for the cortex in the radial and the tangential directions. Furthermore, fitting of the assumed four-parameter linear viscoelastic model with the experimental data resulted in the viscoelastic material parameters.

A triphasic analysis of corneal swelling and hydration control

Physiological studies strongly support the view that hydration control in the cornea is dependent on active ion transport at the corneal endothelium. However, the mechanism by which endothelial ion transport regulates corneal thickness has not been elaborated in detail. In this study, the corneal stroma is modeled as a triphasic material under steady-state conditions. An ion flux boundary condition is developed to represent active transport at the endothelium. The equations are solved in cylindrical coordinates for confined compression and in spherical coordinates to represent an intact cornea. The model provides a mechanism by which active ion transport at the endothelium regulates corneal hydration and provides a basis for explaining the origin of the "imbibition pressure" and stromal "swelling pressure." The model encapsulates the Donnan view of corneal swelling as well as the "pump-leak hypothesis."

The effect of fluid loss on the viscoelastic behavior of the lumbar intervertebral disc in compression

A viscoelastic finite element model of a L2-L3 motion segment was constructed and used to study: (1) the behavior of the intervertebral disc with different amounts of nucleus fluid loss; and (2) the effect of different rates of fluid loss on the viscoelastic behavior of the disc. The results indicate that: (1) The viscoelastic behavior of the intervertebral disc depends to a large extent on the rate of fluid loss from the disc; the intrinsic properties of disc tissue play a role only at the early stage of compressive loading; (2) the axial strain increases, whereas the intradiscal pressure and the posterior radial disc bulge decrease with increasing fluid loss; (3) a decreasing fluid loss rate with a total fluid loss of 10 to 20 percent (from the nucleus) during the first hour of compressive loading best predicts the overall viscoelastic behavior of a disc.

The viscoelastic behavior of the non-degenerate human lumbar nucleus pulposus in shear

The viscoelastic behavior of the nucleus pulposus was determined in shear under transient and dynamic conditions and was modeled using a linear viscoelastic model with a variable amplitude relaxation spectrum. During stress-relaxation tests, the shear stress of the nucleus pulposus relaxed nearly to zero indicative of the fluid nature of the tissue. Under dynamic conditions, however, the nucleus pulposus exhibited predominantly 'solid-like' behavior with values for dynamic modulus (magnitude of G*) ranging from 7 to 20 kPa and loss angle (delta) ranging from 23 to 30 degrees over the range of angular frequencies tested (1-100 rad s-1). This frequency-sensitive viscoelastic behavior is likely to be related to the highly polydisperse populations of nucleus pulposus molecular constituents. The stress-relaxation behavior, which was not linear on a semi-log plot (in the range t1 << t << t2), required a variable amplitude relaxation spectrum capable of describing this frequency sensitive behavior. The stress-relaxation behavior was well described by this linear viscoelastic model with variable amplitude relaxation spectrum; however, the dynamic moduli were underpredicted by the model which may be related to non-linearities in the material behavior.

Finite element models in tissue mechanics and orthopaedic implant design

This article attempts to review the literature on finite element modelling in three areas of biomechanics: (i) analysis of the skeleton, (ii) analysis and design of orthopaedic devices and (iii) analysis of tissue growth, remodelling and degeneration. It is shown that the method applied to bone and soft tissue has allowed researchers to predict the deformations of musculoskeletal structures and to explore biophysical stimuli within tissues at the cellular level. Next, the contribution of finite element modelling to the scientific understanding of joint replacement is reviewed. Finally, it is shown that, by incorporating finite element models into iterative computer procedures, adaptive biological processes can be simulated opening an exciting field of research by allowing scientists to test proposed 'rules' or 'algorithms' for tissue growth, adaptation and degeneration. These algorithms have been used to explore the mechanical basis of processes such as bone remodelling, fracture healing and osteoporosis. RELEVANCE: With faster computers and more reliable software, computer simulation is becoming an important tool of orthopaedic research. Future research programmes will use computer simulation to reduce the reliance on animal experimentation, and to complement clinical trials.